Picard Groups of Moduli Spaces of Torsionfree

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Picard Groups of Moduli Spaces of Torsionfree Sheaves on Curves Usha N. Bhosle Abstract. This article is a short survey of results on the Picard groups of the moduli spaces of semistable torsionfree sheaves on irreducible projective smooth complex curves with at most ordinary nodes as singularities.

1. Introduction In this article we present some old and new results on the Picard groups of the moduli spaces of torsionfree sheaves on curves. X and Y will always denote reduced, irreducible, projective, complex curves. A singular curve will be often denoted by Y and its normalisation by X. Let U (n, d) be the projective moduli space of S-equivalence classes of semistable torsionfree sheaves of rank n, degree d on X and U 0 (n, d) its open subset corresponding to locally free sheaves (vector bundles). The superscript s will denote the open subsets corresponding to stable points. After a short discussion of the case n = 1, we deal with the case n ≥ 2 in the rest of the paper. The first computations of Picard groups of U (n, d), n ≥ 2, were carried out by C.S.Seshadri for n and d coprime and X a smooth curve. We present here a proof by Ramanan [R] expanding on the ideas of Seshadri. These results were extended to the noncoprime case by Drezet and Narasimhan [DN]. Generalizations to singular curves were carried out by us in a series of papers [B3], [B4], [B5]. We state the main results and briefly sketch their proofs. For details, the reader may refer to the original proofs. 2. The rank 1 case The moduli spaces U 0 (1, d) and U (1, d) are respectively the generalized Jacobian d and the compactified Jacobian of degree d of the curve X and are denoted by JX d 0 and J X respectively. JX = JX is the component Pic0 (X) of the Picard group Pic X of X containing the identity. 2000 Mathematics Subject Classification. 14H60, 14D20, 14F05. Key words and phrases. Vector bundles, torsionfree sheaves, moduli spaces, Picard groups. I would like to thank the referee and P.E. Newstead for a very careful reading of the first version and useful suggestions. c

0000 (copyright holder)

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When X is smooth, U (1, 0) = U 0 (1, 0) is the Jacobian JX of X, it is an Abelian variety. Fixing a line bundle L of degree 1 on X, there is the Abel map AL : X → JX , defined by x 7→ L(−x). The induced pull-back map on line bundles gives an isomorphism A∗ : Pic0 (JX ) ∼ = JX L

independent of the choice of L. Thus JX is a selfdual Abelian variety (see [L, Theorem 3, p.156]; [M, Proposition 6.9, p.118]). For a singular curve X, JX is not an Abelian variety, nor is the projective variety J X , the compactified Jacobian of X. The Abel map is defined as the map AL : X → J X , defined by x 7→ Ix ⊗ L, where Ix is the ideal sheaf of x ∈ X. Esteves, Gagn´e and Kleiman have the following generalization of the above result on the selfduality of the Jacobian of a smooth curve [EGK]. Theorem 2.1. (1) Suppose that a singular curve Y has surficial singularities, that is, singularities of embedding dimension 2. Then A∗L : Pic0 (J Y ) → JY has a right inverse independent of the choice of L i.e. there is f : JY → Pic (J Y ) such that A∗L ◦ f = 1JY . A∗L is itself independent of the choice of L. (2) Let Y be a curve with double points as singularities, i.e. with nodes or cusps. Then A∗L is an isomorphism. In fact a relative version of the theorem is proved for flat, projective families of geometrically integral curves over an arbitrary locally Noetherian base scheme [EGK]. The compactified Jacobian of a nodal curve is a seminormal variety. Recall that a variety is called seminormal if all its local rings are seminormal. If B is the normalisation of a ring A, the ring A is said to be seminormal if it contains each b ∈ B such that b2 , b3 ∈ A. The following general result is very useful for computing Picard groups of seminormal varieties. Proposition 2.2. [B5, Proposition 2.5] Let U be a seminormal variety (not necessarily projective) and W the non-normal locus of U , i.e. the scheme of non˜ → U be the normalisation of U and W ˜ the inverse normal points of U . Let π : U ˜ ˜ image of W in U . Define f1 : Pic U → Pic U ⊕ Pic W by f1 (L) = (π ∗ L, L|W ). ˜ ⊕ Pic W → Pic W ˜ by f2 (L1 , L2 ) = L ˜ ⊗ (π ∗ L2 )∗ , for Define f2 : Pic U 1|W ˜ , L2 ∈ Pic W . Then: L1 ∈ Pic U (1) There is an exact sequence f1 f2 ˜ ⊕ Pic W → ˜. Pic U → Pic U Pic W

˜ has p connected (2) Assume further that U is projective, U, W are connected and W components. Then one has an exact sequence f1 f2 ˜ ⊕ Pic W → ˜. 0 → ×p−1 Gm → Pic U → Pic U Pic W

We now apply Proposition 2.2 to compute the Picard groups of JY , the generalized Jacobian, and J Y , the compactified Jacobian, of a nodal curve Y with one node y. J Y is a seminormal projective variety containing the smooth subvariety JY .

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It has a natural desingularisation h : J˜Y → J Y . The map h induces an isomorphism of JY with h−1 (JY ), we identify JY with h−1 (JY ) ⊂ J˜Y . Let X be the normalisation of Y and x, z the points of X lying over y. The variety J˜Y ∼ = P(Px ⊕ Pz ) is a P1 -bundle over JX [AK], Px , Pz being line bundles on JX . Let p0 : J˜Y → JX be the canonical map. J˜Y − JY = D1 ∪ D2 , where Di are divisors such that p0 |Di : Di → JX are isomorphisms. h maps each of D1 and D2 isomorphically onto the smooth Weil divisor J Y − JY . One has ∗ Pic J˜Y ∼ = Pic JX ⊕ Z. = p0 (Pic JX ) ⊕ Z OP(Px ⊕Pz ) (1) ∼

We shall write the operation in Pic additively. Lemma 2.3. [B5, Lemma 3.1] One has Pic JY ∼ = ( Pic JX )/(Pz − Px ), where (Pz − Px ) denotes the subgroup of Pic JX generated by Pz − Px . Proof. Since J˜Y is smooth and JY is open in J˜Y the restriction map Pic J˜Y → Pic JY is surjective. Hence we have an exact sequence 1 → H → Pic J˜Y → Pic JY → 0. Since J˜Y − JY is the union of two irreducible disjoint divisors D1 , D2 the kernel H of the restriction map is generated by O(D1 ) = O(1) − Px , O(D2 ) = O(1) − Pz in Pic JX ⊕ Z ∼ = Pic J˜Y . Let pZ denote the projection to the Z-factor. Then pZ (aO(D1 ) + bO(D2 )) = 0 if and only if a + b = 0. Hence Ker (pZ |H ) is the subgroup generated by (Px − Pz ) in Pic JX . Also pZ |H is surjective (with each generator O(Di ) mapping to 1 ∈ Z). It follows that H∼ = (Px − Pz ) ⊕ Z and Pic JY ∼ = Pic JX /(Px − Pz ). This completes the proof of the lemma.



For u ∈ X let tu denote translation by OX (u) in JX . Denote by HJ the subgroup of Pic JX ⊕ Z defined by HJ = {(L, m) | L ∈ Pic JX , m ∈ Z, t∗z−x L − L = m(Px − Pz )}. Proposition 2.4. [B5, Proposition 3.2] There is an exact sequence 1 → Gm → Pic J Y → HJ → 0. Proof. We sketch the idea of the proof. J Y is a seminormal variety with J Y − JY its non-normal locus. By Proposition 2.2, we have an exact sequence f1 f2 1 → Gm → Pic J Y → Pic J˜Y ⊕ Pic (J Y − JY ) → Pic D1 ⊕ Pic D2 .

We have ∗ Pic J˜Y = p0 (Pic JX ) ⊕ ZO(1),

Pic D1 = (p0 |D1 )∗ Pic JX , Pic D2 = (p0 |D2 )∗ Pic JX . ∗

The map p0 is an injection and (p0 |D1 )∗ , (p0 |D2 )∗ are both isomorphisms. We show that Ker f2 is isomorphic to the subgroup HJ of Pic JX ⊕ Z. 

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Remark 2.5. The connected component Pic0 J Y of Pic J Y containing the identity is isomorphic to JY (Theorem 2.1; [EGK]). Under these identifications, the restriction to Pic0 J Y of the sequence in the statement of Proposition 2.4 is the exact sequence 1 → Gm → JY → JX → 0, where the map JY → JX is the pull-back π ∗ under the normalisation map π. 3. Moduli of vector bundles of rank ≥ 2 Let X be a nonsingular, projective, complex curve of genus g ≥ 2, n, d integers, n ≥ 2. Let UL (n, d) ⊂ UX (n, d) be the subvariety consisting of vector bundles with fixed determinant L. The first result on the Picard groups of moduli spaces of vector bundles of rank ≥ 2 is due to Seshadri. We sketch here a proof of this result given by Ramanan [R] following Seshadri’s ideas. Theorem 3.1. Let (n, d) = 1. Then Pic UL (n, d) ∼ = Z. Proof. (Proof sketch following the proof of [R, Proposition 3.4].) Take F ∈ UL (n, d) and E ∈ UL0 (n − 1, d0 ), d0 an integer. Choose a line bundle M of sufficiently high degree such that there is an injective homomorphism i : E → F ⊗ M , M satisfying certain properties [R, Lemma 3.1]. Note that F ⊗ M/i(E) ∼ = M n ⊗ L ⊗ L0−1 . Then Y := PH 1 (X, M −n ⊗ L0 ⊗ L−1 ⊗ E) parametrizes a family of vector bundles of rank n, determinant L and F occurs in this family. Let Y s ⊂ Y be the subset corresponding to stable bundles in this family, then F ∈ Y s . By the universal property of moduli spaces, there is a morphism λ : Y s → UL (n, d). Since Pic Y → Pic Y s is surjective, rank Pic Y s ≤ 1. Let F be the universal bundle on UL (n, d) × X. Define V = pUL ∗ (Hom(p∗X (E), F ⊗ p∗X (M )) and let Z ⊂ P(V ) be the subset consisting of injective homomorphisms. Then there is an isomorphism Y s → Z and λ factors through Z. Hence ∼ Pic Y s . Pic UL (n, d) → Pic Z = One shows that P(V ) − Z is irreducible [R, Lemma 3.5], so that rank Pic Z + 1 ≥ rank Pic P(V ) = rank Pic UL (n, d) + 1, i.e. rank Pic UL (n, d) ≤ rank Pic Z = rank Pic Y s ≤ 1. Since UL (n, d) is projective, rank Pic UL (n, d) ≥ 1. Hence rank Pic UL (n, d) = rank Pic Z = rank Pic Y s = 1. Then it follows that Pic UL (n, d) ∼ = Pic Z ∼ = Z.



This result was extended to the noncoprime case by Drezet and Narasimhan [DN]. Theorem 3.2. [DN, Theorems A,B and C]. Let X be a smooth, projective curve. (1) Pic UL (n, d) ∼ = Z.

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d (2) Pic UX (n, d) ∼ ⊕ ZLX , where LX is the determinant line bundle = Pic JX on UX (r, d). (3) UX (n, d) and UL (n, d) are locally factorial.

We shall give the proof of Theorem 3.2 as a part of the proof of a more general result on nodal curves. More generally, we have the following result on any irreducible reduced projective curve. Theorem 3.3. [B3, Proposition 2.3]. Let Y be any irreducible projective curve 0 of arithmetic genus gY ≥ 2. Let n ≥ 2 be an integer. Let ULs (n, d) denote the moduli space of stable vector bundles on Y of rank n and determinant isomorphic to a fixed line bundle L. Then 0 Pic ULs (n, d) ∼ = Z or Z/mZ.

Proof. The proof is on similar lines to that of [DN] in the case of a nonsingular curve, one needs some modifications in the singular case. We briefly sketch the proof. By tensoring by a line bundle, we may assume that d >> 0. If E is a semistable vector bundle of high degree and determinant L, then E is globally generated and is given by an exact sequence 0 → OYn−1 → E → L → 0. Let P = P(H 1 (L∗ ⊗ Cn−1 )). There is a family E of vector bundles of rank n, degree d on P × Y . Let Ps = {p ∈ P | Ep stable}. By the universal property of moduli 0 spaces, there is a canonical surjective morphism fE,L : Ps → ULs (n, d). One shows 0 that the induced map Pic ULs (n, d) → Pic Ps (∼ = Z or Z/mZ) is an injection by using an alternative construction of Ps and the construction of the moduli space as follows. 0 The moduli space UYs (n, d) is a GIT quotient of an irreducible nonsingular open subset Rs0 of a suitable quot scheme by P GL(q) (see [N1], Remark, p.167, Chapter 0 (F0 ) is a 5, §7). Let F0 be the universal quotient bundle on Rs0 × Y, F0 := pRs∗ vector bundle. Let Gr0 be the Grassmannian of subspaces V of dimension n − 1 of F0 . There exists a geometric quotient Γ0 = Gr0 /P GL(q) ([DN], 7.3.1). One 0 has Pic Gr0 = Pic Rs0 ⊕ ZOGr0 (1) and hence Pic Γ0 = Pic UYs (n, d) ⊕ Z.OΓ0 (a), where a = gcd(n, d) and OΓ0 (a) is a line bundle on Γ0 ([DN], Proposition 7.2). Let Gr00 ⊂ Gr0 be the open subset corresponding to vector bundle injections OX ⊗V ,→ F0,r , r ∈ Rs0 , V ⊂ H 0 (F)0,r and Gr00 /P GL(q) = Γ00 . Then Γ0 − Γ00 is an irreducible hypersurface [DN, Corollary 7.4]. Computing its ideal sheaf one shows that 0

0 → Pic UYs (n, d) → Pic Γ00 → Z/(d/a)Z → 0 is exact. Let UGr0 be the universal (relative) subbundle on Gr0 . Let T 0 = Isom(Gr0 ⊗ Cn−1 , UGr0 ), T0 = T 0 /GL(q) = P(T 0 )/P GL(q). T0 is an open subset of T where T = P(Hom(Gr0 ⊗ Cn−1 , UGr0 ))/P GL(q), a locally trivial projective bundle over Γ0 . Since the ideal sheaf of T − T0 in T is OT (n − 1) ⊗ p∗Γ0 M , where M is a line bundle on Γ0 , one has 0

0 → Pic Γ00 → Pic T0 → Z/(n − 1)Z → 0

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exact ([DN], 7.8). Let T0L be the inverse image (in T0 ) of UL0 (n, d) ⊂ UY0 (n, d). Then there is an exact sequence 0 d 0 → Pic ULs (n, d) → Pic T0L → Z/(n − 1) Z → 0. a The final observation is that T0L ∼ = PsL ([DN], Proposition 7.9).  3.1. Notation. Henceforth Y denotes an irreducible reduced curve with ordinary nodes as its only singularities (unless otherwise stated) and p : X → Y its normalisation. Let g be the genus of the normalisation X and gY the arithmetic genus of Y . Let UY (n, d) (resp. UY0 (n, d)) be the moduli space of torsionfree semistable sheaves (resp. semistable vector bundles) of rank n and degree d on Y . Let UL0 (n, d) be the closed subset of UY0 (n, d) corresponding to vector bundles with fixed determinant L. UL0 (n, d), UY0 (n, d) are normal quasiprojective varieties. Let 0 0 UYs (n, d) ⊂ UY0 (n, d) and ULs (n, d) ⊂ UL0 (n, d) be the open subsets corresponding to stable vector bundles. For a line bundle L on Y , let UL (n, d) be the closure of UL0 (n, d) in UY (n, d). Theorem 3.4. [B3, Theorem I] Assume that g ≥ 2 and if g = n = 2 then d is odd. Then one has the following. 0 (1) Pic ULs (n, d) ∼ = Z. (2) Pic UL0 (n, d) ∼ =Z. Proof. In view of Theorem 3.3, to prove Part (1) we need only to show that 0 Pic ULs (n, d) has rank ≥ 1. Suppose that Y is nonsingular. Then UL0 (n, d) is projective, hence its Picard group has rank at least 1. Under the assumptions of the theorem, one has 0 codimUL0 (n,d) (UL0 (n, d) − ULs (n, d)) ≥ 2. Since UL0 (n, d) is normal, this implies that 0 0 Pic UL (n, d) ⊂ Pic ULs (n, d) and hence the theorem follows from Theorem 3.3. This is essentially the proof of [DN] in the nonsingular case. If Y is nodal, UL0 (n, d) is not projective, so one has to work harder. Moreover, it is impossible to do the above codimension computations on Y (since tensor products of semistable torsionfree sheaves on Y are not semistable, etc.). Hence we use the normalisation ML (n, d) of UL (n, d) [B1]. ML (n, d) is the moduli space of generalized parabolic bundles of rank n, degree d and determinant isomorphic to L = p∗ (L) on the normalisation X of Y . There is a rational map ML (n, d) → UX,L (n, d), it 0 0 induces an isomorphism MLs (n, d) ∼ = ULs (n, d). Under the assumptions of the theorem, it is proved that the generator of Pic UX,L (n, d) gives a nontorsion element 0 0 0 in Pic MLs (n, d) and hence in Pic ULs (n, d)). Thus Pic ULs (n, d) has rank ≥ 1 [B3, Proposition 2.2]. This proves Part (1). Now if we can show that (3.1)

0

codimUL0 (n,d) (UL0 (n, d) − ULs (n, d)) ≥ 2, 0

0

the normality of UL0 (n, d) will imply that Pic UL (n, d) ⊂ Pic ULs (n, d) and hence 0 0 Pic UL (n, d) has rank 1, proving Part 2 (in fact the restriction map Pic UL (n, d) → 0 Pic ULs (n, d) can be shown to be an isomorphism [B3, Proposition 3.6]). Using s ML (n, d) and the fact that codimUX,L (n,d) (UX,L (n, d) − UX,L (n, d)) ≥ 2 for g ≥ 2 (except for g = n = 2, d even), one can show that the inequality (3.1) holds under these conditions ([B3], Corollary 1.7).

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 Theorem 3.5. [B4, Theorem 3A] With the same assumptions as in Theorem 3.4, let JYd be the generalized Jacobian of Y of degree d i.e. the moduli variety of degree d line bundles on Y . Then 0 (a) Pic UYs (n, d) ∼ = Pic JYd ⊕ Z, 0 ∼ (b) Pic UY (n, d) = Pic JYd ⊕ Z, (c) UY0 (n, d) and UL0 (n, d) are locally factorial. Proof. (a) and (b) are proved using the fact that the determinant map induces an injection on Picard groups and the fibres of the determinant map are isomorphic 0 to UL (n, d). 0 (c) Since ULs (n, d) is smooth, its Picard group and class group coincide. The 0 0 normality of UL (n, d) and the inequality (3.1) imply that the class groups of UL (n, d) 0 0 0 and ULs (n, d) are isomorphic. Moreover, Pic UL (n, d) → Pic ULs (n, d) is injective. 0 Hence it is easy to see that to show that UL (n, d) is locally factorial, it suffices to 0 0 show that the restriction map Pic UL (n, d) → Pic ULs (n, d) is surjective. This is 0 proved using the fact that the determinant line bundle generates Pic ULs (n, d) and 0 comes from Pic UL (n, d) ([B3], p. 262). It is easy to prove the local factoriality of UY0 (n, d) by similar arguments using (a) and (b).  4. Moduli of torsionfree sheaves on nodal curves. We start with the simple case of curves with lower genera. For low genera and ranks, there are explicit descriptions of moduli spaces of torsionfree sheaves. This makes the computation of their Picard groups much easier. 4.1. Case gY ≤ 2. . Any irreducible nonsingular (respectively nodal) curve Y with gY = 1 is a nonsingular (respectively nodal) Weierstrass curve. If Y is nonsingular, it is an elliptic curve. If Y is nodal, it has normalisation P1 and a unique singular point which is an ordinary node. If gcd(n, d) = h, the moduli space U (n, d) is isomorphic to S h (Y ), the hth symmetric product of Y (see [At], [Tu] for the nonsingular case; [BBH, Corollary 6.47], [HLST, Remark 1.27] for the singular case). In particular, for (n, d) = 1, UY (n, d) is isomorphic to Y . For Y nodal, UY0 (n, d) is isomorphic to the variety of nonsingular points of Y i.e. to the affine variety A1 − 0. Proposition 4.1. Let gY = 1. (i) If (n, d) = 1, then Pic UY (n, d) ∼ = Pic Y . If Y is singular, then Pic UY (n, d) ∼ = Gm ⊕ Z and Pic UY0 (n, d), Pic UL (n, d) are trivial. (ii) If h = (n, d) > 1 then Pic UY (n, d) ∼ = Pic S h (Y ). ∼ For L ∈ JY , Pic UL (2, 0) = Z. If Y is singular, then Pic UL0 (2, 0) and Pic UY0 (2, 0) are trivial. Proof. By the explicit description, Pic UY (n, d) ∼ = Pic S h (Y ). If (n, d) = 1, then UL (n, d) is a point, so its Picard group is trivial. If Y is nodal, then Pic Y ∼ = Gm ⊕ Pic P1 ∼ = Gm ⊕ Z

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and since UY0 (n, d) ∼ = A1 − 0, it has a trivial Picard group. For the results in rank 2 note that for L = O, every E ∈ UL (2, 0) is S-equivalent to N ⊕ N ∗ for N ∈ J Y . Hence UL (2, 0) is isomorphic to the quotient of Y by Z/2Z which is isomorphic to P1 . Moreover, UL0 (2, 0) ∼ = A1 and UY0 (2, 0) is fibred over 1 0 ∼ JY = A − 0 with fibres UL (2, 0). For more details, see [B4, Proposition 2.3].  For gY = 2, Y is a hyperelliptic curve. If Y is singular, then Y has one or two nodes, the normalisation is an elliptic curve or a projective line. Suppose that d is odd. If Y is nonsingular, then UL (2, d) is isomorphic to a nonsingular intersection Q of two quadrics ([N], [NRa, Theorem 4]). If Y is nodal, then the study of the extension of the determinant map U 0 (2, d) → JYd to UY (2, d) [B1] shows that the singular set UL (2, d) − UL0 (2, d) consists of direct images of stable vector bundles of rank 2 and a suitable fixed determinant on the (partial) normalisations of Y . Note that the (partial) normalisations are curves of arithmetic genus 1 and the projective line. By Grothendieck’s theorem, there are no stable vector bundles on a projective line. On a curve of arithmetic genus 1, there is a unique stable vector bundle of rank 2 and a fixed determinant of odd degree. It follows that UL (2, d) − UL0 (2, d) consists of one or two singular points according as Y has one or two nodes. If d is even, one has UL0 (2, d) = UL (2, d) ∼ = P3 and the subset UL (2, d)−ULs (2, d) is isomorphic to the Kummer variety in P3 corresponding to Q considered as a quadratic complex of lines in P3 ([NRa, Theorem 2], [B2, Corollary 3.5]). Lemma 4.2. The Kummer variety K is an irreducible quartic surface in P3 . Proof. The curve Y is associated to a pencil of quadrics in k 6 with the following standard equation (k = C in our case, in general k can be an algebraically closed field of characteristic 6= 2). q1

=

I X i=1

q2

=

I X i=1

2Xi Yi +

X

Xj2

j≥I+1

(λi Xi Yi + Xi2 ) +

X

λj Xj2

j≥I+1

Here I = 0, 1 or 2 according as Y is nonsingular, has one node or two nodes. The intersection of quadrics can be considered as a quadratic complex of lines in P3 of type [111111], [21111], [2211] in the three cases. For the first type, the singular surface K is the classical Kummer surface, which is an irreducible quartic surface with 16 nodes. In the other two cases, the equation of K may be found, for instance, in Article 271 on p.211 and article 186 on p.219 of [J]; in both cases K is a quartic which is easily checked to be irreducible.  Proposition 4.3. Let gY = 2. Then (i) Pic UL0 (2, d) = Pic UL (2, d) ∼ = Z for d even. 0 (ii) Pic ULs (2, d) ∼ = Z/4Z for d even. 0 (iii) Pic UL0 (2, d) = Pic ULs (2, d) ∼ = Z for d odd. (iv) Pic UY0 (2, d) ∼ = Pic JY ⊕ Z for d odd. (v) Pic UY (2, d) ∼ = Pic JY ⊕ Z for d even and Y nonsingular. Proof. (i) follows immediately from the fact UL0 (2, d) = UL (2, d) ∼ = P3 . 0 3 ∼ (ii) Since UL (2, d) = P is nonsingular there is a surjective homomorphism of Picard

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0

groups Pic UL0 (2, d) → Pic ULs (2, d). As the complement of ULs (2, d) in UL0 (2, d) is a divisor of the line bundle OP3 (4) (by Lemma 4.2), the kernel of the surjective 0 homomorphism is 4Z. Hence Pic ULs (2, d) ∼ = Z/4Z for d even. (iii) In case Y is nonsingular, Q is a nonsingular complete intersection in P5 of dimension 3. By [H, Corollary 3.2, p.179], Pic Q ∼ = ZOQ (1), proving the result. Alternatively, the result also follows from Theorem 3.3 as ULs (2, d) = UL (2, d) is a projective variety. ˜L (2, d) → UL (2, d) be a normalisation. Since Suppose that Y is nodal. Let p : U UL0 (2, d) is normal (in fact nonsingular), p is an isomorphism over UL0 (2, d). As the singular set UL (2, d) − UL0 (2, d) consists of one or two points, it follows that ˜L (2, d) − p−1 UL0 (2, d) = codim UL (2, d) − UL0 (2, d) ≥ 3. codim U ˜L (2, d) is normal, this implies that Since U ˜L (2, d) ,→ Pic p−1 UL0 (2, d) ∼ Pic U = Pic UL0 (2, d). ˜L (2, d) and hence Since UL (2, d) is projective, so is U ˜L (2, d) ≥ 1 . rank Pic UL0 (2, d) ≥ rank Pic U Now it follows from Theorem 3.3 that Pic UL0 (2, d) ∼ = Z. (iv) This follows from (iii) as in the following Part(v) (or as in [B4, Theorem 3A]). (v) The morphism det : UY (2, d) = UY0 (2, d) → JYd is a P3 fibration (not locally trivial). Hence for any line bundle M on JYd , one has det∗ (det∗ M ) = M , so that det∗ is injective on line bundles. Let N be a line bundle on UY (2, d). The restriction of N to the fibre over L is O(mL ), mL ∈ Z. JYd being irreducible, mL = m must be constant. The allowable values of m form a non-trivial subgroup of Z, so restriction to a fibre determines a surjection Pic UY (2, d) → Z. By the seesaw theorem, the kernel of this surjection is isomorphic to Pic JYd . This completes the proof.  4.2. Case gY ≥ 3. . In the rest of this section, we assume that Y is a nodal curve of genus gY ≥ 3 with one node y and x, z points of the normalisation X lying over y. Our aim is to compute the Picard groups of UY (2, d), UL (2, d) and UL0 (2, d). For gY ≥ 3, the variety UY (2, d) is not normal, it is seminormal [NR]. Theorem 4.4. [B4, Proposition 2.8] For gY ≥ 3, one has: 0 (1) ULs (2, d) ∼ = Z. (2) UL0 (2, d) ∼ = Z. 0 0 ˜L → UL be a Proof. We write ULs = ULs (2, d), UL0 = UL0 (2, d). Let p : U normalisation, it is an isomorphism over UL0 (since UL0 is normal). Then codim ˜L − p−1 U 0 = codim UL − U 0 ≥ 3 [B4, Lemma 2.5]. Since U ˜L is normal, this U L L −1 0 0 ∼ ˜ implies that Pic UL ,→ Pic (p UL ) = Pic UL . Since UL is projective, so is ˜L and hence rank (Pic U ˜L ) ≥ 1, rank (Pic U 0 ) ≥ 1. Since U 0 is normal and U L L 0 0 codim (UL0 − ULs ) ≥ 3 [B4, Lemma 2.4], we have Pic UL0 ,→ Pic ULs . Thus rank 0 0 (Pic ULs ) ≥ 1. Now it follows from Theorem 3.3 that Pic ULs ∼ = Z and hence Pic UL0 ∼  = Z.

The variety UY (2, d) has a filtration U := UY (2, d) = W2 ⊃ W1 ⊃ W0

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where Wi are seminormal, the non-normal locus of Wi is Wi−1 for i = 1, 2 and W0 ∼ = UX (2, d − 2) is smooth. Let U 0 = W2 − W1 , U1 = W1 − W0 , U0 = W0 be the strata (for more details, see [NR] or [B5, section 4.2]). Denote by U1,L (2, d) the subvariety of U1 (2, d) corresponding to torsionfree sheaves with a fixed (not locally free) determinant [B4, 2.10]. Theorem 4.5. [B4, Theorem 2 and Theorem 3B] Let gY ≥ 2 and let L be a torsionfree sheaf of rank 1 on Y which is not locally free. Then (a) Pic U1,L (2, d) ∼ = Z. Moreover, assume that d is odd if gY = 2. Then s (b) Pic U1,L (2, d) ∼ = Z, Pic U1s (2, d) ∼ = Pic JX ⊕ Z. ∼ (c) Pic U1 (2, d) = Pic JX ⊕ Z. (d) U1 (2, d) is locally factorial. Proof. This is proved as in the case of UL0 (2, d), U 0 (2, d).



To compute the Picard group of U using our Proposition 2.2, we need to know the Picard group of the normalisation P of U and that of the non-normal locus W1 . The normalisation P was constructed by Narasimhan and Ramadas [NR] as the moduli space of GPS (generalized parabolic sheaves) on the normalisation X of Y . A GPS is a sheaf E, which is torsionfree outside {x, z} ⊂ X, together with a 2-dimensional quotient Q of Ex ⊕ Ez . A GPS is called a GPB (generalized parabolic bundle) if E is a vector bundle. The moduli space P is a normal projective (irreducible) variety with rational singularities. It has a filtration P = P 2 ⊃ P 1 ⊃ P 0. We have P 1 = D1 ∪ D2`where D `1 ,D2 are irreducible and normal Cartier divisors [NR]. P 0 = (D1 ∩ D2 ) D1 (0) D2 (0) where Di (0) are closed subsets of Di , i = 1, 2. Let P 0 = P − P 1 , P1 = P 1 − P 0 , P0 = P 0 . The normalisation map p : P → U maps P i onto Wi . Its restriction to P 0 is an isomorphism onto U 0 . Moreover, for each j, the restriction p |Dj : Dj → W1 is a normalisation map with D0 j = Dj − (D1 ∩ D2 ) − Dj (0) mapping isomorphically onto U1 = W1 − W0 . Theorem 4.6. [B5, Theorem I] For gX ≥ 2, rank 2 and odd degree d, one has the following. (1) P − P0 is nonsingular. (2) Pic (P − P0 ) ∼ = Pic UX (2, d) ⊕ Z. d (3) Pic P ∼ ⊕ Zp∗ LY , = Pic JX LY being the determinant line bundle on U and p : P → U the normalisation map. The divisor class group of P (and of P − P0 ) is isomorphic to Pic UX (2, d) ⊕ Z. Idea of Proof. Part(1) is proved by using the smooth determinant map [B1] detp : P − P0 → J˜Y and proving that all its fibres are smooth [B5, Proposition 4.6].

PICARD GROUPS OF MODULI SPACES OF TORSIONFREE SHEAVES ON CURVES

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For (2), let E → UX (2, d) × X be the universal bundle. Define Ex = E|UX (2,d)×x , Ez = E|UX (2,d)×z and let pr : (G := Gr(Ex ⊕ Ez )) → UX (2, d) be the Grassmannian bundle of two dimensional quotients of Ex ⊕ Ez . Then G parametrizes a family of GPBs giving a morphism G s → P . One shows that it induces an isomorphism from a big open subset of G onto a big open subset of P − P0 , here big means the complement has codimension ≥ 2 [B5, Proposition 4.6]. The proof of (3) is far more complicated [B5, Proposition 4.14]. Since P is normal and P0 is of codimension ≥ 2, Pic P ,→ Pic(P − P0 ), but computation of the image is hard. We show that d Pic Di ∼ ⊕Z∼ = Pic JX = Pic Di (0)

and study the restriction maps from Pic P to these schemes.



Using the fact that W1 is seminormal with normalisation D1 and non-normal locus W0 , by applying Proposition 2.2, we compute the Picard group of W1 (for odd degree d). We partially succeed in computing the Picard group of U similarly. Theorem 4.7. [B5, Theorem II]. ∼ Pic JX ⊕ Z. (1) Pic W0 = (2) Pic W1 is given by an exact sequence 0 1 → Gm → Pic W1 → HW ⊕Z→0 1 0 d where HW is isomorphic to the subgroup G of Pic JX defined by 1 d G = {L ∈ Pic JX | (⊗OX (z − x))∗ L = L}.

(3) One has an exact sequence 1 → Gm → Pic U

resW1



Pic W1 .

Proof. (1) is proved using the isomorphism W0 ∼ = UX (2, d − 2) [B5, Corollary 4.11]. (2) is proved in [B5, Proposition 4.15] using (1) and Proposition 2.2. (3) is deduced from Proposition 2.2 using (2) and a computation of Pic D1 [B5, Proposition 4.17].  Remark 4.8. Computing the image HU of Pic U in Pic W1 explicitly seems difficult as we do not have enough information about Pic P1 . One has HU = {B ∈ Pic W1 | p∗ (B) = (Det∗ L + m LY ) |P1 , L ∈ G, m ∈ Z}. Here LY = p∗ LY , LY being the determinant line bundle on Y . Det denotes the d determinant morphism P → JX which associates to a GPB (E, Q) the determinant of the underlying vector bundle E. Note that every GPS is S-equivalent to a GPB, so there is a well-defined morphism Det [Su].

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References [AK]

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